Weighted omega-Restricted One Counter Automata

Abstract

Let $S$ be a complete star-omega semiring and $\Sigma$ be an alphabet. For a weighted $\omega$-restricted one-counter automaton $\mathcal{C}$ with set of states $\{1,\dots ,n\}$, $n\ge 1$, we show that there exists a mixed algebraic system over a complete semiring-semimodule pair $((S\ll {\Sigma }^{\ast }\gg {)}^{n\times n},(S\ll {\Sigma }^{\omega }\gg {)}^n)$ such that the behavior $\Vert \mathcal{C}\Vert$ of $\mathcal{C}$ is a component of a solution of this system. In case the basic semiring is $\mathbb{B}$ or ${\mathbb{N}}^{\infty }$ we show that there exists a mixed context-free grammar that generates $\Vert \mathcal{C}\Vert$. The construction of the mixed context-free grammar from $\mathcal{C}$ is a generalization of the well-known triple construction in case of restricted one-counter automata and is called now triple-pair construction for $\omega$-restricted one-counter automata.